$τ_d$-tilting theory for linear Nakayama algebras
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910819730587648 |
|---|---|
| author | Rundsveen, Endre S. Vaso, Laertis |
| author_facet | Rundsveen, Endre S. Vaso, Laertis |
| contents | Support $τ$-tilting pairs, functorially finite torsion classes and $2$-term silting complexes are three much studied concepts in the representation theory of finite-dimensional algebras, which moreover turn out to be connected via work of Adachi, Iyama and Reiten. We investigate their higher-dimensional analogues via $τ_d$-rigid pairs, $d$-torsion classes and $(d+1)$-term silting complexes as well as the connections between these three concepts. Our work is done in the setting of truncated linear Nakayama algebras $Λ(n,l)=\mathbf{k} \mathbb{A}_{n}/\mathrm{rad}{\mathbf{k} \mathbb{A}_{n}}^l$ admitting a $d$-cluster tilting module. More specifically, we classify $τ_d$-rigid pairs $(M,P)$ of $Λ(n,l)$ with $|M|+|P|=n$ via an explicit combinatorial description and show that they can be characterized by a certain maximality condition as well as by giving rise to a $(d+1)$-term silting complex in $\mathrm{K}^b(\mathrm{proj}(Λ(n,l)))$. We also describe all $d$-torsion classes of $Λ(n,l)$. Finally, we compare our results to the classical case $d=1$ and investigate mutation with a special emphasis on the case where $d$ equals the global dimension of $Λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19505 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $τ_d$-tilting theory for linear Nakayama algebras Rundsveen, Endre S. Vaso, Laertis Representation Theory 16G20, 16S90, 18E10, 18G99 Support $τ$-tilting pairs, functorially finite torsion classes and $2$-term silting complexes are three much studied concepts in the representation theory of finite-dimensional algebras, which moreover turn out to be connected via work of Adachi, Iyama and Reiten. We investigate their higher-dimensional analogues via $τ_d$-rigid pairs, $d$-torsion classes and $(d+1)$-term silting complexes as well as the connections between these three concepts. Our work is done in the setting of truncated linear Nakayama algebras $Λ(n,l)=\mathbf{k} \mathbb{A}_{n}/\mathrm{rad}{\mathbf{k} \mathbb{A}_{n}}^l$ admitting a $d$-cluster tilting module. More specifically, we classify $τ_d$-rigid pairs $(M,P)$ of $Λ(n,l)$ with $|M|+|P|=n$ via an explicit combinatorial description and show that they can be characterized by a certain maximality condition as well as by giving rise to a $(d+1)$-term silting complex in $\mathrm{K}^b(\mathrm{proj}(Λ(n,l)))$. We also describe all $d$-torsion classes of $Λ(n,l)$. Finally, we compare our results to the classical case $d=1$ and investigate mutation with a special emphasis on the case where $d$ equals the global dimension of $Λ$. |
| title | $τ_d$-tilting theory for linear Nakayama algebras |
| topic | Representation Theory 16G20, 16S90, 18E10, 18G99 |
| url | https://arxiv.org/abs/2410.19505 |